×

zbMATH — the first resource for mathematics

Bernstein functions of several semigroup generators on Banach spaces under bounded perturbations, II. (English) Zbl 06905103
Summary: The paper deals with multidimensional Bochner-Phillips functional calculus. In the previous paper by the author bounded perturbations of Bernstein functions of several commuting semigroup generators on Banach spaces where considered, conditions for Lipschitzness and estimates for the norm of commutators of such functions where proved. Also in the one dimensional case the Frechet differentiability of Bernstein functions of semigroup generators on Banach spaces where proved and a generalization of Livschits-Kreǐn trace formula derived. The aim of the present paper is to prove the Frechet differentiability of operator Bernstein functions and the Livschits-Kreǐn trace formula in the multidimensional setting.

MSC:
47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47L20 Operator ideals
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] A. B. ALEKSANDROV ANDV. V. PELLER, Operator Lipschitz functions, Russian Mathematical Surveys 71 (2016), no. 4, 605-702. · Zbl 1356.26002
[2] A. B. ALEKSANDROV ANDV. V. PELLER, Krein’s trace formula for unitary operators and operator Lipschitz functions, Funct. Anal and Appl. 50 (2016), no. 3, 167-175. · Zbl 1450.47009
[3] C. BERG, K. BOYADZHIEV ANDR.DELAUBENFELS,Generation of generators of holomorphic semigroups, J. Austral. Math. Soc. (Series A) 55 (1993), 246-269. · Zbl 0796.47009
[4] CH. BERG, J. P. R. CHRISTENSEN, P. RESSEL, Harmonic analysis on semigroups, Grad. Texts in Math., vol.100, Springer-Verlag, New York-Berlin, 1984. · Zbl 0619.43001
[5] M. S. BIRMAN ANDD. R. YAFAEV, The spectral shift function. The papers of M. G. Kre˘ın and their further development, Algebra i Analiz 4 (1992), 1-44 (Russian). English transl.: St. Petersburg Math. J. 4 (1993), 833 - 870.
[6] S. BOCHNER, Harmonic analysis and the theory of probabylity, University of California Press, Berkeley and Los Angeles, 1955.
[7] N. BOURBAKI, Elements de mathematique. Livre VI. Integration. 2nd ed., Ch. 1 - 9, Hermann, Paris, 1965-1969.
[8] YU. L. DALETSKII ANDS. G. KRE˘IN, Integration and differentiation of functions of Hermitian operators and application to the theory of perturbations (Russian), Trudy Sem. Functsion. Anal., Voronezh. Gos. Univ. 1 (1956), 81-105.
[9] A. DEFANT ANDK. FLORET, Tensor norms and operator ideals, North-Holland, Amsterdam, 1993. · Zbl 0774.46018
[10] E. HILLE ANDR. PHILLIPS, Functional Analysis and Semigroups, Amer. Math. Soc., Providence, R.I., 1957.
[11] E. KISSIN, D. POTAPOV, V. SHULMAN,ANDF. SUKOCHEV, Operator smoothness in Schatten norms for functions of several variables: Lipschitz conditions, differentiability and unbounded derivations, Proc. London Math. Soc. 105, (2012), no. 4, 661-702. · Zbl 1258.47022
[12] M. G. KRE˘IN, On a trace formula in perturbation theory, Mat. Sbornik 33 (1953), 597-626 (Russian). · Zbl 0052.12303
[13] I. M. LIFSHITZ, On a problem in perturbation theory connected with quantum statistics, Uspekhi Mat. Nauk 7 (1952), 171-180 (Russian).
[14] M. MALAMUD, H. NEIDHART, Trace formulas for additive and non-additive perturbations, Adv. in Math. 274 (2015), 736-832. · Zbl 1380.47011
[15] M. MALAMUD, H. NEIDHART, V. PELLER, Analytic operator Lipschitz functions in the disc and a trace formula for functions of contractions, Functional Analysis and its Applications, 51 (2017), no. 3, 33-55, Preprint, arXiv:1705.07225 v1 [math. FA].
[16] M. MALAMUD, H. NEIDHARDT, V. PELLER, A trace formula for functions of contractions and analytic operator Lipschitz functions, Comptes Rendus Acad. Sci. Paris, Ser. I, 355 (2017), 806-811. · Zbl 1381.47008
[17] A. R. MIROTIN, Bernstein functions of several semigroup generators on Banach spaces under bounded perturbations, Operators and Matrices 11 (2017), 199-217. · Zbl 1367.47023
[18] A. R. MIROTIN, On some functional calculus of closed operators on Banach space. III. Some topics in perturbation theory, Izvestiya VUZ. Matematika 12 (2017), 24-34 (Russian); English transl.: Russian Math. 12, to appear.
[19] A. R. MIROTIN, On theT -calculus of generators for C0-semigroups, Sib. Matem. Zh., 39 (1998), no. 3, 571-582; English transl.: Sib. Math. J. 39 (1998), no. 3, 493-503. · Zbl 0949.47033
[20] A. R. MIROTIN, Criteria for Analyticity of Subordinate Semigroups, Semigroup Forum 78 (2009), no. 2, 262-275. · Zbl 1162.47036
[21] A. R. MIROTIN, Functions from the Schoenberg classT on the cone of dissipative elements of a Banach algebra, Mat. Zametki 61 (1997), no. 4, 630-633; English transl.: Math. Notes 61 (1997), no. 3-4, 524-527.
[22] A. R. MIROTIN, Functions from the Schoenberg classT act in the cone of dissipative elements of a Banach algebra, II, Mat. Zametki 64 (1998), no. 3, 423-430; English transl.: Math. Notes 64 (1998), no. 3-4, 364-370.
[23] A. R. MIROTIN, MultidimensionalT -calculus for generators of C0semigroups, Algebra i Analiz 11 (1999), no. 2, 142-170; English transl.: St. Petersburg Math. J. 11 (1999), no. 2, 315-335.
[24] A. R. MIROTIN, On some properties of the multidimensional Bochner-Phillips functional calculus, Sib. Mat. Zhurnal, 52 (2011), no. 6, 1300 - 1312; English transl.: Siberian Mathematical Journal 52 (2011), no. 6, 1032-1041. · Zbl 1237.47016
[25] A. R. MIROTIN, On joint spectra of families of unbounded operators, Izvestiya RAN: Ser. Mat. 79 (2015), no. 6, 145-170; English transl.: Izvestiya: Mathematics 79 (2015), no. 6, 1235-1259.
[26] A. R. MIROTIN, Properties of Bernstein functions of several complex variables, Mat. Zametki, 93 (2013), no. 2, 257-265; English transl.: Math. Notes 93 (2013), no. 2. · Zbl 1262.32009
[27] A. R. MIROTIN, On multidimensional Bochner-Phillips functional calculus, Probl. Fiz. Mat. Tekh. 1 (2009), (2013)1, 63-66 (Russian). · Zbl 1262.47022
[28] O. P. MISRA, J. L. LAVOINE, Transform analysis of generalized functions, North Holland, Amsterdam, 1986. · Zbl 0583.46032
[29] J. N. PANDEX, On the Stieltjes transform of generalized functions, Proc. Camb. Phil. Soc. 71 (1972), (2013)1, 85-96.
[30] R. S. PATHAK, A distributional generalized Stieltjes transformation, Proc. Edinburgh Math. Soc. 20 (1976), no. 1, 15-22. · Zbl 0334.44004
[31] V. V. PELLER, The behavior of functions of operators under perturbations. A glimpse at Hilbert space operators, 287-324, Oper. Theory Adv. Appl. 207, Birkhauser Verlag, Basel, 2010. Preprint, http://arxiv.org/abs/0904.1761 [math. FA].
[32] V. V. PELLER, The Lifshitz-Krein trace formula and operator Lipschitz functions, Proc. Amer. Math. Soc. 144 (2016), 5207-5215, doi: http://dx.doi.org/10.1090/proc/13140. · Zbl 1353.47023
[33] J. ROZENDAAL, F. SUKOCHEV,ANDA. TOMSKOVA, Operator Lipschitz functions on Banach spaces, Studia Mathematica 232 (2016), no. 1, 57-92. · Zbl 1372.47022
[34] R. SHILLING, R. SONG, Z. VONDRACEK, Bernstein functions. Theory and applications, de Greyter, Berlin-New York, 2010.
[35] D. R. YAFAEV, Mathematical scattering theory, volume 105 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1992. · Zbl 0761.47001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.